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1answer
112 views

Simplifying an expression using tools from Fourier transform

Can anyone simplify the following expression? I guess something from Fourier transform can help: $f(\omega) = \lim_\limits{R \to \infty} \frac{1}{R^2} \int_{r=0}^{R}{re^{ \omega r^{-\gamma}}} ...
0
votes
0answers
55 views

Finite Ramanujan expansions over a finite field

I'm wondering if we could have an analogy in finite fields. After all, the Discrete Fourier Transform (DFT) has been generalized to finite fields as well (with essentially identical properties as in ...
4
votes
1answer
269 views

Roots of characteristic function of “reciprocal gamma measure”

Let us call a measure $\mu$ on the Borel $\sigma$-algebra $\mathfrak{B}_{(0,\infty)}$ of subsets of $(0,\infty)$ a reciprocal gamma measure if it is absolutely continuous with respect to the Lebesgue ...
2
votes
0answers
59 views

Generalization of Pitt's theorem

Pitt's theorem (Pitt 1937), states that the one-dimensional Fourier tranform is well defined and continuous between the weighted spaces $L^p(\mathbb{R},|x|^{bp}dx)$ and $L^q(\mathbb{R},|x|^{\beta ...
0
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0answers
42 views

Relating the R-Transform in Free Probability to noncommutative group representations

In traditional (commutative) probability theory, sums of random variables correspond to convolution of distribution functions, which plays well with the Fourier Transform. In free (noncommutative) ...
3
votes
0answers
109 views

Isbell duality in Joyal and Street's Introduction to Tannaka Duality

In Sec. 3 of Joyal and Street's Introduction to Tannaka Duality and Quantum Groups, the authors give a commutative triangle of isomorphisms of compact topological groups (Corollary 8). This diagram ...
0
votes
1answer
103 views

Fourier series and transform related to Epicycles

Let $\gamma:\mathbb{R}\to\mathbb{C}$ be a continuous periodic curve having a bounded variation. 1) Is it true that one can find a sequence of numbers $(r_n)_{n\in\mathbb{N}}\subset (0,\infty)$ and ...
0
votes
0answers
49 views

Fourier transform of the realization of a stationary process in the space of tempered distributions?

A path of a stationary sequence of random variables $y_t$ does not have a discrete-time Fourier transform in the classical sense because it is not summable. This leads to considering the spectral ...
5
votes
1answer
124 views

How are the infinity norm of Fourier transforms of sign vectors distributed?

This is a follow up to an earlier resolved question. Define the $n$-dimensional discrete Fourier transform via the matrix $$ D_{s,t} := \omega^{st}, $$ where $\omega=\exp(-2\pi i/n)$. Notice that $D$ ...
7
votes
1answer
1k views

Eliminating Gibbs phenomenon, and approximating with jumping functions in Fourier Analysis : An attempt and a question in this regard

Physical Motivation : Hear to these audio files S_f and P_f. S_f is Fourier partial sum and P_f is the new reconstruction, both use spectrum only in the region (0,4KHz) for reconstructing the ...
5
votes
0answers
79 views

Real interpolation space between the Wiener algebra and $L^2$

The Wiener algebra $W_n$ is the image by the Fourier transform of $L^1(\mathbb R^n)$. What is the (complex) interpolation space between $W_n$ and $L^2(\mathbb R^n)$? It is probably not true that for ...
0
votes
0answers
95 views

Inverse Fourier transform of $\frac{1}{\sqrt{\xi_1} + \xi_2}$

Consider the inverse Fourier transform of $\frac{1}{\sqrt{\xi_1} + \xi_2}$. My question is, how can we conclude about the decay properties, support and smoothness of the inverse Fourier transform? I ...
1
vote
0answers
107 views

Technical question about a Fourier transform

I would like to know if there is an explicit expression for the Fourier transform of the following function: $$f(x)=\mathbb{1}_{(0,\infty)}e^{-x-ix^2},$$ or to know where I can find some techniques to ...
0
votes
0answers
117 views

Problem with operator and Fourier transform

I am currently dealing with a problem in functional analysis where I want to show the following. Let $\phi_{k+1}(x):=\sqrt{2} \sum_{n \in \mathbb{Z}} h(n) \phi_k(2x-n)$ and $\phi_0= \chi_{[0,1]},$ if ...
0
votes
0answers
34 views

Is Wiener amalgam spaces $W^{2,1}(\mathbb R)\subset C_0(\mathbb R)$?

I have been learning Wiener amalgam spaces. In Wiener amalgam spaces $W(X, L^2)$, I am taking $X=\mathcal{F}L^{1}=\{f\in L^{\infty}(\mathbb R):\hat{f}\in L^{1}\},$ and $m(x)=1.$ Take $f(x)= ...
1
vote
1answer
161 views

Finite trigonometric polynomial

I noticed by numerical and some explicit calculations for a few examples that for real-valued finitely supported functions $\phi \in L^2(\mathbb{R})$ we have that $T(x):= \sum_{n \in \mathbb{Z}} ...
1
vote
2answers
136 views

Proof of Biot-Savart theorem in 3D using distributions and fourier transform

While studying the book Vorticity and Incompressible Flow by A. Majda and A. Bertozzi I came across the following Biot-Savart theorem in 3D: The solution to: $$div \ v=0$$ $$curl(v)=\omega$$ is ...
3
votes
1answer
123 views

Characterizations of Wiener algebra

The Wiener algebra $\mathcal W$ is defined as $\text{Fourier}(L^1(\mathbb R))$, i.e. the image by the Fourier transform of $L^1(\mathbb R)$. Riemann-Lebesgue's lemma ensures that $$ \mathcal W\subset ...
1
vote
0answers
46 views

Can we choose $g$ so that $\|(g\widehat{(f^{3})})^{\vee}\|_{L^{p}} \leq C \|g_{1}f\|_{L^{2}}^{r} \|(g_{2}\hat{f})^{\vee}\|_{L^{s}}$?

[I have asked this question on S.E. M; but I have not got any answer; and hope this is o.k. for M.O] Let $f, f^{2}, f^{3}\in L^{q}(\mathbb R)\cap C_{0}(\mathbb R)$ where $ q\geq p, \ \text{and}$ and ...
2
votes
0answers
97 views

On uniform or simple convergence of Poisson Summation formula

Under good conditions on an even function $f(x)$ we have the Poisson Summation formula ($x>0$): $$f(0) + 2 \sum\limits_{n =1}^{\infty} f(nx)= \frac{1}{x} \left( \hat{f}(0) + 2 \sum\limits_{n ...
3
votes
0answers
83 views

Can we extend the twisted Poisson Summation formula with functions having a singularity in zero?

The following "twisted" Poisson Summation formula for $\chi$ primitive of conductor $q$ : $$ \sum_{n\in\mathbb{Z}}\chi(n)f\left(\frac{nx}{\sqrt{q}}\right) = ...
3
votes
2answers
308 views

Fourier transform of compactly supported distribution is smooth

My advisor made the comment that if $u\in \mathcal{E}'$ is a compactly supported distribution, then $\hat{u}(\xi)\in C^{\infty}(\mathbb{R}^n)$ is actually a smooth function (not merely a distribution ...
1
vote
1answer
152 views

Is there a Poisson Summation formula for imprimitive Dirichlet characters?

I was wondering if there exists a Poisson Summation formula (like the one existing with primitive character) for imprimitive Dirichlet characters ? For a primitive Dirichlet character $\chi$ we have: ...
0
votes
0answers
305 views

What is the Fourier transform of this function?

Consider the function $$ f(x_1,x_2)=|x_1x_2|^{-\alpha/2}\int_{\mathbb{R}} \frac{e^{it(x_1+u)}-1}{i(x_1+u)} \frac{e^{it(x_2-u)}-1}{i(x_2-u)} |u|^{-\beta}du. $$ It is known that $f(x_1,x_2)\in ...
5
votes
0answers
121 views

Are these identities Newton series?

Newton series is the following expansion of a function: $$f(x)=\sum_{k=0}^\infty \binom{x}k \Delta^k [f]\left (0\right)=\sum_{n=0}^{\infty} {x\choose n} \sum_{k=0}^n{n\choose k}(-1)^{k-n}f(k)$$ Now ...
7
votes
2answers
505 views

Newton series and Fourier transform - is there an analogy?

Fourier expansion for a function: $$f(x)=\frac{1}{2\pi}\int_{-\infty}^{+\infty} e^{- i \omega x}\int_{-\infty}^{+\infty}e^{i\omega t}f(t)dt \, d\omega$$ Newton series expansion of a function: ...
2
votes
1answer
257 views

Connection between the Fourier transform of f and |f|

If $f\in L^p(R)$ with $1\leq p\leq 2$, then Hausdorff-Young inequality implies that the Fourier transform $\widehat{f}\in L^{p'}$, $p'$ is the dual exponent of $p$, and $$ ...
1
vote
0answers
33 views

Multidimensional Filters

Say you want to design a LP FIR filter with low pass cutoff $fc$, transition band $fc$ to $fs$ and ripple factor $dp$ at passband and $ds$ at stop band. If one divides the frequencies by $\pi$, then ...
1
vote
0answers
161 views

Fourier transform of a matrix represented compact lie group

In physics, I come across this kind of integration (in the nonlinear sigma model): \begin{equation} S[g] = \frac{1}{\lambda} \int d^dr\ \text{tr}[\triangledown g\triangledown g^{-1}] \end{equation} ...
0
votes
1answer
130 views

A kind of Discrete Fourier Transform

Given a $z\in \mathbb{C}^N$, the DFT of $z$ is given for every $k\in [0,N-1]_\mathbb{N}$ by $$DFT_z(k)=\frac{1}{N} \sum_{j=0}^{N-1} z_j\, \omega^{-k j}$$ where I have denoted by $\omega$ the $N$-th ...
4
votes
3answers
238 views

Is there a compactly supported function that its Fourier transfrom vanishes at given n real points?

My question is as follows: Given ${{\lambda }_{1}},\,{{\lambda }_{2}},...,{{\lambda }_{n}}\in \mathbb{R}$ where $\underset{1\le j\le n-1}{\mathop{\min }}\,\left| {{\lambda }_{j+1}}-{{\lambda }_{j}} ...
2
votes
0answers
64 views

request for any expository works in pointwise convergence of double Fourier series and especially a paper by Hardy

Quart. J. Math. Volume 37, Issue 1, Pages 53-79 On double Fourier series, and especially those which represent the double zeta-function with real and incommensurable parameters. Hardy, G.H. I am not ...
0
votes
0answers
102 views

A question on a Fourier transform

(This may sound too simple a question to be asked in MO. But it is a research problem but not an exercise, so I post it here. If it is no appropriate, I will move it to Stackexchange.) Let $0 < a ...
5
votes
1answer
163 views

Largest area of a compactly supported positive definite function

Consider a continuous function $f: \mathbb{R} \rightarrow \mathbb{R}$, supported on $[-1,1]$, of positive type. Assume $f(0) = 1$; what is the "largest area" $\int f\,dx$ that can be achieved? To be ...
-3
votes
1answer
489 views

A question about pointwise convergence of Fourier transform in $N$-dimensions

I am retreating back on this statement, after some explorations and calculation Bow to Willie and others who were skeptical on this. Main difficulty can be seen in this reference. But I must mention ...
6
votes
2answers
312 views

Reverse Hausdorff Young for nonnegative functions

The classical Hausdorff-Young inequality states that $$ \Vert \widehat{f} \Vert_{p'} \leq \Vert f \Vert_p \text{ for } 1 \leq p \leq 2. $$ For $p=2$, we even have equality due to Plancherel. If we ...
1
vote
0answers
83 views

Inversion of Fourier transform of a multivariate gamma distribution in polar form?

Let $\mathbb{S}^{N-1}$ be the unit sphere in $\mathbb{R}^N$ under the Euclidean norm $||\cdot||$. Let $\mu$ be an infinitely divisible Borel measure. If there exists a finite measure $\alpha$ on ...
2
votes
0answers
115 views

Discrete versus Continuous Hilbert Transform

Let me define the Fourier transform of a function $u$, say in the Schwartz space $\mathscr S(\mathbb R)$ as $ \hat u(\xi)=\int_{\mathbb R} e^{-2iπ x\cdot \xi} u(x) dx. $ The Hilbert transform ...
1
vote
1answer
146 views

Fourier approximation error in L^2 for piecewise continuous functions

Let $u:[0,2\pi)\to \mathbb{R}$ be the step function $$u(x) = \begin{cases} 1 & \text{if } x \in [0,\pi), \\ 0 & \text{if } x \in [\pi,2\pi) \end{cases}$$ By a direct computation, one ...
18
votes
1answer
441 views

Functions $f$ on $\mathbb{Z}/N\mathbb{Z}$ with $|f|$ and $|\widehat{f}|$ constant

Let $N$ be a positive integer; for simplicity I'm happy to assume it's an odd prime but I'm interested in the general case too. Let $f \colon \mathbb{Z}/N\mathbb{Z} \to \mathbb{C}$ and let ...
5
votes
2answers
233 views

Eigenstates of Fourier transformation

Let $\gamma$ be defined on $\mathbb R^n$ by $\gamma (x)=e^{-π x^2}$. With $\mathcal F$ standing for the Fourier transformation defined on the Schwartz space by $$ (\mathcal F u)(\xi)=\int e^{-2iπ ...
3
votes
1answer
170 views

A example on Fourier tranform of a continous compactly supported function

I am trying to find a continuous compactly supported function $f$ such that the Fourier transform $f^{ft}$ and derivative $(f^{ft})'$ of the $f^{ft}$ decay faster than exponential rates, that is ...
4
votes
2answers
126 views

Inverse Fourier of $\omega^{-1+{\rm i}\alpha} u(\omega-1)$

Let $\alpha$ be an arbitrary real number and define \begin{align} \widehat{f}(\omega)=\left\{\begin{array}{ll} \omega^{-1+{\rm i}\alpha}, & \omega>1,\\ 0, & \textrm{otherwise}. \end{array} ...
3
votes
2answers
257 views

How do functions operate in a Sobolev space $H^{s}$?

Let $s>\frac{1}{2};$ and define a Sobolev space as follows: $$H^{s}(\mathbb R)=\{f\in L^{2}(\mathbb R):[\int_{\mathbb R} |\hat{f}(\xi)|^{2}(1+|\xi|^{2})^{s}d\xi]^{1/2}<\infty \}.$$ Fact: Let $m$ ...
1
vote
1answer
105 views

Does the Fourier transform of a non-strictly positive real kernel $\Phi(t)$ always generate entire function with complex zeros? [closed]

Question (1) Does the Fourier transform of a non-strictly positive real kernel $f(t)$ always generate an entire function $g(z)$ with complex zeros? $$g(z)=\int_{-\infty}^{\infty}f(t) ...
1
vote
0answers
71 views

Cross-correlation of two functions which are not fixed

I am trying to cross-correlate two functions, but one of which is changing each 'step' of the cross-correlation. I want to cross-correlate T(f) and ...
0
votes
1answer
169 views

$\widehat{f\ast g}= \hat{f} \cdot \hat{g}$ for $f, \hat{f} \in L^{p}(\mathbb R)\cap C(\mathbb R) (1<p<\infty, p\neq 2), g\in \mathcal{S}(\mathbb R)$? [closed]

(This may be very simple question for MO; I had post it to math stack exchange few days back but I could not get any answer(or comment) to it) It is well-known that, for $f,g \in L^{1}(\mathbb R).$ ...
0
votes
0answers
100 views

Mellin transform of time-shifted function

The Mellin transform of a function $f(x)$ can be written as $$ \mathcal M[f(x);z]=\int_0^\infty f(x)x^{z-1} dx $$ Is there a simple expression for the Mellin transform of the function $f(x-x_0)$? ...
4
votes
0answers
76 views

Szegő curve for partial sum of Taylor series of Riemann $\Xi(z)$ function

I am sorry that this is long post. But it might be of interest to you. This post is related to zeros of partial sum of Taylor series of $e^x-1$. Entire functions $e^z$, $\cos(z)$, and $\sin(z)$ can ...
3
votes
2answers
357 views

Are there any new results on approximating Riemann $\Xi$ function by Polya-like Fourier transforms?

I posted [this question][1] at math.stackexchange.com and was told that it is more appropriate to post this research related question here at mathoverflow. So I re-post it below. Riemann $\Xi(z)$ ...